(i) The first term \(a = \sin^2 \theta\) and the second term \(a + d = \sin^2 \theta \cos^2 \theta\).

The common difference \(d = \sin^2 \theta \cos^2 \theta - \sin^2 \theta\).

Factor out \(\sin^2 \theta\):

\(d = \sin^2 \theta (\cos^2 \theta - 1)\).

Since \(\cos^2 \theta - 1 = -\sin^2 \theta\), we have:

\(d = -\sin^4 \theta\).

(ii) The sum of the first 16 terms of an arithmetic progression is given by:

\(S_{16} = \frac{16}{2} [2a + 15d]\).

With \(\theta = \frac{1}{3} \pi\), \(\sin \theta = \frac{\sqrt{3}}{2}\), so \(a = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}\).

\(d = -\left(\frac{\sqrt{3}}{2}\right)^4 = -\frac{9}{16}\).

Substitute \(a\) and \(d\) into the sum formula:

\(S_{16} = 8 \left[2 \times \frac{3}{4} + 15 \times -\frac{9}{16}\right]\).

\(S_{16} = 8 \left[\frac{3}{2} - \frac{135}{16}\right]\).

\(S_{16} = 8 \left[\frac{24}{16} - \frac{135}{16}\right]\).

\(S_{16} = 8 \times -\frac{111}{16}\).

\(S_{16} = -\frac{888}{16} = -\frac{55}{2}\).